To put the present invention into context, a description of the zenithal bistable device of WO 97/14990 will be given with reference to FIGS. 1 and 2 wherein a liquid crystal layer (1) is in contact with an asymmetric surface alignment grating (1) of pitch W and groove depth h. The liquid crystal director is denoted by the vector n. The contour lines (3) are perpendicular to orientation of the liquid crystal director (n) and the grating surface has been treated, for example by coating with a homeotropic surfactant (not shown), to induce local homeotropic alignment of the liquid crystal at the interface (4) between the liquid crystal layer (2) and the surface alignment grating (1).
The term homeotropic takes the meaning, well known to a person skilled in the art, that the director (n) is oriented substantially perpendicularly to the local surface and herein a surface alignment grating is taken to mean an array of repeating elements of a unit length, as would be understood by a person skilled in the art. Herein the term “same azimuthal plane” is explained as follows: let the walls of a cell lie in the x, z plane, which means the normal to the cell walls is the y axis. Two pretilt angle in the same azimuthal plane means two different molecular positions in the same x, y plane. Herein, as defined in FIG. 1, the term pretilt shall mean the tilt of the director away from the x-z plane at some distance away from the surface where the director is invariant in the x-z plane; hence perfectly planar alignment gives zero pretilt and perfectly homeotropic alignment gives 90° pretilt.
It was demonstrated in a particular embodiment of WO 97/14990 that a nematic liquid crystal in contact with an alignment grating surface coated with a homeotropic surfactant can adopt either a non-defect (FIG. 1a) or defect (FIG. 1b) alignment configuration. Liquid crystal defects can, in simple terms, be considered as a local area of discontinuity in the director field and are well known to persons skilled in the art. A summary of the theory relating to liquid crystal defects and disclinations can be found in P. G. deGennes, “The physics of liquid crystals” (Clarendon press, 1974).
In the non-defect structure of FIG. 1a the nematic liquid crystal will, at the interface (4) between the surface alignment grating (1) and the liquid crystal (3), orient so as to be substantially perpendicular to the local surface of the grating (1). Within a short distance of the grating-liquid crystal interface (4), compared with the overall thickness of the liquid crystal cell, the liquid crystal will adopt a homeotropic alignment configuration of θp≈90°.
In the defect structure of FIG. 1b, so-called defects of strength −½ (5) will form in the vicinity of concave defect sites (7) and so-called defects of strength +½ (6) will form in the vicinity of convex defect sites (8). The result of the formation of the +½ and −½ defect pair is that within a short distance of the grating-liquid crystal interface (4), as compared with the overall thickness of the liquid crystal cell, the nematic liquid crystal will adopt a configuration of a pretilt (in this example θp≈45°) lower than that formed for the non-defect structure of FIG. 1a. Note that defects can only occur in pairs, and each pair must be of an equal and opposite magnitude. A more complete explanation of defects of strength +½ and −½ can be found in P. G. deGennes, “The physics of liquid crystals” (Clarendon press, 1974), and would be known by a person skilled in the art.
Herein, defects of strength of approximately −½ (within a range of −1 to 0) and +½ (within a range of 0 to 1) are termed the “−½ defect” and the “+½ defect” respectively. The positions on a grating surface where +½ and −½ defects form are associated with regions of convex and concave minimum radii of curvature of the grating surface. The terms “−½ defect site” and “+½ defect site”, are taken to mean the region on a grating surface where a person skilled in the art would reasonably expect either a −½ or +½ defect to form using common general knowledge, documents such as P. G. deGennes, “The physics of liquid crystals” (Clarendon press, 1974), and the teachings contained hereinafter. The term “defect site” means either a −½ defect site or a +½ defect site and “defect sites” is simply more than one such defect site.
A suitable cell configuration to exploit the existence of the bistable surface described with respect to FIG. 1, is shown in cross section in a stylised form in FIG. 2. The cell configuration of FIG. 2 comprises a layer of nematic liquid crystal material with a positive dielectric anisotropy (2) sandwiched between a first glass wall (9) and a second glass wall (10). The first glass cell wall (10) is treated, for example by coating with lecithin (not shown), to induce homeotropic alignment of the nematic liquid crystal at the glass cell wall and liquid crystal interface (11). The second glass cell wall (10) is coated with a bistable surface alignment grating (1), the profile of which is as described with respect to FIG. 1.
The device of FIG. 2 allows the liquid crystal molecules to adopt either of two stable configurations, as shown in FIGS. 2a and 2b. 
In FIG. 2a, homeotropic alignment of the liquid crystal is induced at the first substantially flat cell wall surface (9) because of the homeotropic treatment (not shown). At the second cell wall (10) the liquid crystal (2) adopts a non-defect state (as described with respect to FIG. 1a) and induces homeotropic alignment of the liquid crystal (2) within a short distance, with respect to the overall thickness of the liquid crystal cell (d), of the grating-liquid crystal interface (4). A uniform homeotropic (high tilt) alignment of the liquid crystal is obtained throughout the bulk of the cell.
In FIG. 2b, homeotropic alignment of the liquid crystal is obtained at the first substantially flat cell wall surface (9) because of the homeotropic treatment (not shown). At the second cell wall (10) the surface alignment grating (1) adopts a defect state (as described with respect to FIG. 1b) and induces low tilt alignment (in this example θp≈45°) of the liquid crystal (2) within a short distance, with respect to the overall thickness of the liquid crystal cell (d), of the grating-liquid crystal interface (4). A splayed liquid crystal structure is thus formed. For many nematic materials, a splay or bend deformation will lead to a macroscopic flexoelectric polarisation, which is represented by the vector P in FIG. 2. A dc pulse is used to couple to this polarisation and depending on its sign will either favour or disfavour the configuration of FIG. 2(b). The application of pulses of positive and negative sign can be used to drive the system between the two stable states.
Simple Model of Grating Induced Surface Pretilt
The pretilt associated with a particular pair of +½ and −½ defects has been found to depend on the relative positions of the +½ and −½ defects per grating period. A model of the pretilt induced at a surface by a grating structure will now be described with reference to FIGS. 1 to 3.
Consider a nematic liquid crystal restricted in a configuration such that at all points the director of the liquid crystal is parallel to the x-y plane. Let θ(x, y) be the tilt angle between the director at (x,y) and the x-axis. Then, in an untwisted configuration, θ(x, y) completely specifies the director field.
The static director field is governed by a torque balance equation obtained from minimising the Frank-Oseen free energy of a nematic liquid crystal. Under planar configuration and taking an approximation that the splay and bend elastic constants are equal, both the Frank-Oseen free energy expression and the corresponding torque balance equation can be reduced to a simple form as:
                    G        =                              K            2                    ⁢                      ∫                                                  ⁢                                          ⅆ                x                            ⁢                              ⅆ                y                            ⁢                              {                                                                            (                                                                        ∂                          θ                                                                          ∂                          x                                                                    )                                        2                                    +                                                            (                                                                        ∂                          θ                                                                          ∂                          y                                                                    )                                        2                                                  }                                                                        (        1        )                                0        =                              (                                                            ∂                  2                                                  ∂                                      x                    2                                                              +                                                ∂                  2                                                  ∂                                      y                    2                                                                        )                    ⁢          θ                                    (        2        )            where G is the Frank-Oseen free energy per unit length and K is the bend or splay elastic constant. The torque balance equation 2 is simply the Laplace equation. It is well-known in complex variables that any analytic function is a solution to the Laplace equation. In particular, if we seek solutions for a configuration of disclination lines normal to the x-y plane, the problem is directly analogous to the potential flow fluid mechanics or two dimensional electrostatics; see for example M. R. Speigel, “theory and problems of complex variables” (Schaum, New York, 1964) and P. M. Morse and H. Feshbach, “Methods of theoretical physics” (McGraw-Hill, New York, 1953). The equivalents of the disclination lines are the line sources or sinks in potential flow fluid mechanics and line charges in two dimensional electrostatics.
For an isolated disclination core at the origin and with the boundary condition that the directors are un-anchored at infinity, solution to the Laplace equation 2 is given by:
                    θ        =                                            M              2                        ⁢                        ⁢                          {                              ln                ⁡                                  (                                      x                    +                    iy                                    )                                            }                                +          α                                    (        3        )                                          =                                                    M                2                            ⁢                                                tan                                      -                    1                                                  ⁡                                  (                                      y                    x                                    )                                                      +            α                          ⁢                                                      (        4        )            where  denotes the imaginary part of a complex function, M/2 is the strength of the disclination and the angle, α, is an arbitrary constant of integration. m is an integer necessary to ensure that the director orientation is preserved in any close circuit which enclosed the disclination core at the origin. Equation 3 is well-known as the stream line function in potential flow fluid mechanics or the flux line function in two dimensional electrostatics.
Having obtained the tilt function of 2 for the director field, the free energy 1 can be readily evaluated, see for example P. G. deGennes & J Prost, “The physics of liquid crystals” (Academic, New York, 1993), to yield:
                    G        =                                            K              ⁢                                                          ⁢              π                        4                    ⁢                      M            2                    ⁢                      ln            ⁡                          (                              R                λ                            )                                                          (        5        )            where R is the dimension of the system and λ is the disclination core radius, and is typically the order of a few molecular lengths (circa 100 Å). Note that the free energy, G, diverges logarithmically with the system size.
If a configuration of parallel disclination lines with their cores pinned at the points (a1, b1), . . . , (an, bn) . . . in the x-y plane is considered, then the tilt angle θ(x, y) which specifies the director field is the superposition of the contributions from each individual disclination lines is:
                              θ          =                                                    ∑                n                            ⁢                                                θ                  n                                ⁡                                  (                                      x                    ,                    y                                    )                                                      +            α                          ⁢                                  ⁢                                  ⁢        where                            (        6        )                                          θ          n                =                                            M              n                        2                    ⁢                    ⁢                      {                          ln              ⁡                              (                                  x                  -                                      a                    n                                    +                                      i                    ⁡                                          (                                              y                        -                                                  b                          n                                                                    )                                                                      )                                      }                                              (        7        )                                          =                                                    M                n                            2                        ⁢                                          tan                                  -                  1                                            ⁡                              (                                                      y                    -                                          b                      n                                                                            x                    -                                          a                      n                                                                      )                                                    ⁢                                                      (        8        )            θn defines the director field due an isolated disclination line of strength Mn/2 pinned at (an,bn). Similarly, the free energy, G, is also the sum of all individual contributions:
                              G          =                                    K              2                        ⁢                          ∫                                                ⅆ                  x                                ⁢                                  ⅆ                  y                                ⁢                                                                                                                        ∑                        n                                            ⁢                                              (                                                                                                            ∂                                                              θ                                n                                                                                                                    ∂                              x                                                                                ,                                                                                    ∂                                                              θ                                n                                                                                                                    ∂                              y                                                                                                      )                                                                                                  2                                                                    ⁢                                                      (        9        )                                =                                                            K                ⁢                                                                  ⁢                π                            4                        ⁢                                          ∑                n                            ⁢                                                M                  n                  2                                ⁢                                  ln                  ⁡                                      (                                          R                      λ                                        )                                                                                +                                    K              2                        ⁢                                          ∑                                  n                  ≠                                      n                    ′                                                              ⁢                                                ∫                                      r                                          n                      ⁢                                                                                          ⁢                                              n                        ′                                                                              R                                ⁢                                                      {                                          θ                      n                                        }                                    ⁢                                      (                                                                  M                                                  n                          ′                                                                    2                                        )                                    ⁢                                                                          ⁢                                                            ⅆ                      r                                        r                                                                                                          (        10        )                                          =                                                                      K                  ⁢                                                                          ⁢                  π                                4                            ⁢                                                (                                                            ∑                      n                                        ⁢                                          M                      n                                                        )                                2                            ⁢                              ln                ⁡                                  (                                      R                    λ                                    )                                                      +                                                            K                  ⁢                                                                          ⁢                  π                                4                            ⁢                                                ∑                                      n                    ≠                                          n                      ′                                                                      ⁢                                                      M                    n                                    ⁢                                      M                                          n                      ′                                                        ⁢                                      ln                    ⁡                                          (                                              R                                                  r                                                      n                            ⁢                                                                                                                  ⁢                                                          n                              ′                                                                                                                          )                                                                                                          ⁢                                  ⁢        where                            (        11        )                                                      r                          n              ⁢                                                          ⁢                              n                ′                                              =                                                                (                                                      a                    n                                    ,                                      b                    n                                                  )                            -                              (                                                      a                                          n                      ′                                                        ,                                      b                                          n                      ′                                                                      )                                                                ⁢                                                      (        12        )            
Equation 3 is a generalisation of the case of a pair of disclination lines as described in, for example, P. G. deGennes & J Prost, “The physics of liquid crystals” (Academic, New York, 1993). The integral in equation 10 is obtained by an integration by parts technique. The integral is performed along a cut from (an,bn) in the direction of (an,bn)–(an′,bn′). Along the cut, θn goes through a discontinuous change of 2πMn as one traverses across the cut in the anti-clockwise direction. Evaluated the integral 10 yields equation 11 for the free energy. As noted in P. M. Chaikin & T. C. Lubensky, “Principles of condensed matter physics” (Cambridge University Press, 1995), the 1nR divergence in part of equation 11 can be eliminated if the sum of the strengths of all disclination lines in the system is zero.
Consider a simple case in which an infinite array of disclination lines each with strength ±M/2 are pinned at regular interval of L at positions ±(a−nL, b) where n is an integer in the range [−∞, ∞]. Let z=x+iy and c=a+ib. By using equation 6, one obtains the tilt angle function for the director field:
                    θ        =                                            M              2                        ⁢                                          ∑                                  n                  =                                      -                    ∞                                                  ∞                            ⁢                                                          ⁢                                              ⁢                                  {                                      ln                    ⁡                                          (                                              z                        -                        c                        -                        nL                                            )                                                        }                                                              -                                    M              2                        ⁢                                          ∑                                  n                  =                                      -                    ∞                                                  ∞                            ⁢                                                          ⁢                                              ⁢                                  {                                      ln                    ⁡                                          (                                              z                        +                        c                        -                        nL                                            )                                                        }                                                              +          α                                    (        13        )                                          =                                                    M                2                            ⁢                            ⁢                              {                                  ln                  ⁡                                      [                                                                                            z                          -                          c                                                                          z                          +                          c                                                                    ⁢                                                                        ∏                                                      n                            =                            1                                                    ∞                                                ⁢                                                                              1                            -                                                                                          (                                                                                                      z                                    -                                    c                                                                    nL                                                                )                                                            2                                                                                                            1                            -                                                                                          (                                                                                                      z                                    +                                    c                                                                    nL                                                                )                                                            2                                                                                                                                            ]                                                  }                                      +            α                          ⁢                                                      (        14        )                                          =                                                    M                2                            ⁢                            ⁢                              {                                  ln                  ⁡                                      [                                                                  sin                        ⁢                                                                                                  ⁢                                                  π                          L                                                ⁢                                                  (                                                      z                            -                            c                                                    )                                                                                            sin                        ⁢                                                                                                  ⁢                                                  π                          L                                                ⁢                                                  (                                                      z                            +                            c                                                    )                                                                                      ]                                                  }                                      +            α                          ⁢                                                      (        15        )            
Equation 15 was obtained from equation 14 by using the infinite products expansion of the sine function as in, for example, P. M. Morse and H. Feshbach, “Methods of theoretical physics” (McGraw-Hill, New York, 1953). In regions far from the array,
                                          lim                          y              →                              ±                ∞                                              ⁢          θ                =                                            ±                              M                2                                      ⁢                          {                                                                    tan                                          -                      1                                                        ⁡                                      [                                                                  tan                        ⁢                                                  π                          L                                                ⁢                                                  (                                                      x                            -                            a                                                    )                                                                                            tanh                        ⁢                                                  π                          L                                                ⁢                                                                            y                                                                                                                ]                                                  -                                                      tan                                          -                      1                                                        ⁡                                      [                                                                  tan                        ⁢                                                  π                          L                                                ⁢                                                  (                                                      x                            +                            a                                                    )                                                                                            tanh                        ⁢                                                  π                          L                                                ⁢                                                                            y                                                                                                                ]                                                              }                                +          α                                    (        16        )                                          =                                                    ±                                  M                  2                                            ⁢                              {                                  π                  ⁢                                                                          ⁢                                                            2                      ⁢                      a                                        L                                                  }                                      +            α                          ⁢                                                      (        17        )            
From equation 17 it can be seen that the grating will induce zero pretilt when a =L/2. Thus for a symmetrical sinusoidal grating, as described in WO 97/14990, +½ and −½ defects will form along the grating equally spaced by the distance a (which is half the grating pitch length L). This defect configuration produces a zero pretilt defect state.
Referring to FIG. 3, if within one period of the grating there is only one position of maximum curvature which is convex and only one which is concave and the zero pretilt condition a=L/2 is not fulfilled (in this case because the grating is asymmetric) the alignment grating will induce a finite surface tilt. For the asymmetric grating structure of FIG. 3, +½ defects will form at the +½ defect sites (21) and −½ defects will form at the −½ defect sites (22) and the surface alignment grating will impart a pretilt to the liquid crystal, within a short distance of the surface compared to the overall cell thickness, according to equation 17.
From the model described above it can been seen that it is the position of defect pair formation, and not the degree of symmetry of the grating surface, which determines the surface pretilt that is imparted to a liquid crystal layer by a surface alignment grating at a short distance from the grating surface.
Detailed Model of Grating Induced Surface Pretilt.
A more rigorous analytical model of the liquid crystal pretilt induced by the formation of defects on a surface alignment grating will now be described.
Theoretical examples derived from the model will then be described with reference to FIGS. 4 and 5.
Surface deformation of a nematic liquid crystal due to the shape of a surface interface have been investigated. Only interfaces with a periodic profile and homeotropic surface alignment of the liquid crystal are considered. For smooth surfaces with no kink, surface line defects with integral π topology nucleated on the interface surface can have full translational freedom on the surface. The model described hereinafter analyses the energetics of the line surface defects. Under equal elastic approximation and assumed planar director field, the only relevant field variable for the director field is the tilt angle θ of the director and the governing static equation is the Laplace equation. By means of a conformal mapping techniques, θ and the surface deformation energy can be obtained without the small amplitude approximation used by Berreman (W. D. Berreman, Phys. Rev. Lett. 28, 1683 (1972)) and deGennes (P. G. deGennes, “Physics of liquid crystals”, Oxford, 77 (1974)). Unlike Barbero (G. Barbero, Lett Nuovo Cimento, 29, 553 (1980)), only smooth surfaces are considered and hence there is no surface defect pining.
Let w=u+iv be the conformal co-ordinates of the transformed z=x+iy co-ordinates such that v=0 corresponds to the surface profile y=η(x). Consider an array of surface line defects of alternating strength ±mπ at the positions λ±+2πn where n=(−∞, ∞) on the v=0 line. The total surface deformation energy per period per unit length for a surface with a periodic maximum and minimum energy is:
                    g        =                              K                          2              ⁢              L                                ⁢                                    ∫              0              ∞                        ⁢                                                  ⁢                                          ⅆ                v                            ⁢                                                ∫                  0                  L                                ⁢                                                                  ⁢                                                      ⅆ                    u                                    ⁢                                      {                                                                                                                        ∂                            ρ                                                                                ∂                            u                                                                          ⁢                                                                              ∂                            θ                                                                                ∂                            v                                                                                              -                                                                                                    ∂                            ρ                                                                                ∂                            v                                                                          ⁢                                                                              ∂                            θ                                                                                ∂                            u                                                                                                                }                                                                                                          (        18        )                                          ≈                                    K                              2                ⁢                L                                      ⁢            π            ⁢                          {                                                [                                                            max                      ⁡                                              (                        ρ                        )                                                              -                                          min                      ⁡                                              (                        ρ                        )                                                                              ]                                +                                                                          m                                                        ⁡                                      [                                                                  ρ                        ⁡                                                  (                                                      λ                            +                                                    )                                                                    -                                              ρ                        ⁡                                                  (                                                      λ                            -                                                    )                                                                                      ]                                                  +                                                                                                  m                                                              2                                    ⁢                                      ln                    ⁡                                          [                                                                                                    sin                            2                                                    ⁡                                                      (                                                                                          (                                                                                                      λ                                    +                                                                    -                                                                      λ                                    -                                                                                                  )                                                            /                              2                                                        )                                                                                                                                sin                            2                                                    ⁡                                                      (                                                          a                              /                              2                                                        )                                                                                              ]                                                                                  }                                      ⁢                                                      (        19        )            where ρ is the harmonic conjugate of θ and a is the radius of the defect line core measured in w. The first square bracket term of equation 19 is the nematic deformation energy without any defect. The middle square bracket term of equation 19 is due to the coupling of the defect and non-defect deformation. The last term of equation 19 depends only on the defects. With g, the energetic stability of the defect can be analysed with respect to the variation of λ± and m. The director configuration is specified by the tilt angle:
                              ρ          +                      i            ⁢                                                  ⁢            θ                          =                              ln            ⁢                                          ⅆ                w                                            ⅆ                z                                              +                                    ∑                              α                =                ±                                      ⁢                          α              ⁢                                              m                                            ⁢              ln              ⁢                                                          ⁢                              sin                ⁡                                  (                                                            w                      -                                              λ                        α                                                              2                                    )                                                              +                      i            ⁢                                                  ⁢                          θ              0                                                          (        20        )            where θ0 is the alignment angle of the director at the interface (i.e. local) surface.
In the limit v=∞, hence
                                          lim                          v              →              ∞                                ⁢          θ                =                                            ∑                              α                ±                                      ⁢                          α              ⁢                                              m                                            ⁢              Im              ⁢                              {                                  ln                  [                                                                          ⁢                                      sin                    ⁡                                          (                                                                        w                          -                                                      λ                            α                                                                          2                                            )                                                        ]                                }                                              +                      θ            0                                              (        21        )                                          =                                                    -                                                    m                                                              ⁢                              (                                                                            λ                      +                                        -                                          λ                      -                                                        2                                )                                      ⁢                                                  +                          θ              0                                      ⁢                                                      (        22        )            
If a periodic surface profile, with a multitude of maxima and minima within a single period, is considered the surface energy, g, is generalised to:
                    g        =                              K                          2              ⁢              L                                ⁢                      π            ⁡                          (                                                g                  0                                +                                  g                  od                                +                                  g                  dd                                            )                                                          (        23        )            where g0, god and gdd correspond to the surface energies due to the purely non-defect field, non-defect-defect field coupling and defect-anti-defect field coupling, where:
                                              ⁢                              g            o                    ≈                                    ∑                                                i                  max                                ,                                  j                  min                                                      ⁢                          (                                                ρ                  ⁢                                                                          ⁢                                      i                    max                                                  -                                  ρ                  ⁢                                                                          ⁢                                      j                    min                                                              )                                      ⁢                                                      (        24        )                                                          ⁢                              g            od                    =                                    ∑              α                        ⁢                                          m                α                            ⁢                              ρ                ⁡                                  (                                      λ                    α                                    )                                                                    ⁢                                                      (        25        )                                          g          dd                =                              ∑                          α              ,              β                                ⁢                                    m              α                        ⁢                          m              β                        ⁢                          ln              ⁡                              [                                                      sin                    2                                    ⁡                                      (                                                                                            λ                          α                                                -                                                  λ                          β                                                                    2                                        )                                                  ]                                                                        (        26        )            
The equations above are only valid for
                    ∑        α            ⁢              m        α              =    0    ,otherwise g will increase linearly with the system height. Notice that go and gdd are always positive. Hence, the condition for the possibility of the existence of multi-stable states is if:
                                          g            od                    +                      g            dd                          =                                            ∑              α                        ⁢                                          m                α                            ⁢                              {                                                      ρ                    ⁡                                          (                                              λ                        α                                            )                                                        +                                                            ∑                      β                                        ⁢                                                                  m                        β                                            ⁢                      ln                      ⁢                                                                                          ⁢                                                                        sin                          2                                                ⁡                                                  (                                                                                                                    λ                                α                                                            -                                                              λ                                β                                                                                      2                                                    )                                                                                                                    }                                              ≤          0                                    (        27        )            for some combinatorial permutation of m's and λ's.
The director field configuration is given by:
                              ρ          +                      i            ⁢                                                  ⁢            θ                          =                              ln            ⁢                                          ⅆ                w                                            ⅆ                z                                              +                                    ∑              α                        ⁢                                          m                α                            ⁢              ln              ⁢                                                          ⁢                              sin                ⁡                                  (                                                            w                      -                                              λ                        α                                                              2                                    )                                                              +                      i            ⁢                                                  ⁢                          θ              o                                                          (        28        )            and the far from the surface alignment grating and liquid crystal interface the surface behaviour of θ is:
                                          lim                          v              →              ∞                                ⁢          θ                =                              -                                          ∑                α                            ⁢                                                m                  α                                ⁡                                  (                                                            λ                      α                                        2                                    )                                                              +                      θ            o                                              (        29        )            
Under planar (zero twist) configuration and one elastic constant approximation, the zero volt nematoelastic can be realised as the 2 dimensional Laplace equation:
                                                                        ∂                2                            ⁢              θ                                      ∂                              x                2                                              +                                                    ∂                2                            ⁢              θ                                      ∂                              y                2                                                    =        0                            (        30        )            where θ is the tilt angle of the nematic director field defined in the x-y plane. The problem is analogous to the two dimensional electrostatic, where θ is the electrostatic potential.
The solution to equation 30 is:
                              θ          ⁡                      (            w            )                          =                              Im            ⁢                                                  ⁢                          ln              ⁡                              (                                                      ⅆ                    z                                                        ⅆ                    w                                                  )                                              +                      π            2                                              (        31        )            where z=(x+iy) and w=(u+iv) are complex planes related by a conformal mapping such that z(w)=x(w)+iy(w) is the grating surface profile when w=u+i0.
Solution 31 gives the homeotropic alignment at the grating surface:
                                                                        θ                ⁡                                  (                                      w                    =                                          u                      +                                              0                        ⁢                        i                                                                              )                                            =                            ⁢                                                Im                  ⁢                                                                          ⁢                                      ln                    ⁡                                          (                                                                                                    ⅆ                            x                                                                                ⅆ                            u                                                                          +                                                  i                          ⁢                                                                                    ⅆ                              y                                                                                      ⅆ                              u                                                                                                                          )                                                                      +                                  π                  2                                                                                                        =                            ⁢                                                                    tan                                                                  -                        1                                            ⁢                                                                                                                            ⁡                                      (                                                                                            ⅆ                          y                                                                          ⅆ                          u                                                                    /                                                                        ⅆ                          x                                                                          ⅆ                          u                                                                                      )                                                  +                                  π                  2                                                                                        (        32        )            
The conformal mapping technique, z→ω, is required to be analytic in the upper half plane of ω for θ and hence the energy to be non-singular. Therefore,limv→∞w=cz  (33)
If c is real then:
                                          lim                          v              →              ∞                                ⁢          θ                =                                                            lim                                  v                  →                  ∞                                            ⁢                              Im                ⁡                                  (                                                            ⅆ                      z                                                              ⅆ                      w                                                        )                                                      +                          π              2                                =                      π            2                                              (        34        )            
From the solution of equation 31, one obtains the conjugate function, ρ, of θ which also satisfies the Laplace equation 30
                              ρ          ⁡                      (            w            )                          =                              Re            ⁢                                                  ⁢                          ln              ⁡                              (                                                      ⅆ                    z                                                        ⅆ                    w                                                  )                                              =                      ln            ⁢                                                                          ⅆ                  z                                                  ⅆ                  w                                                                                                      (        35        )            ρ and θ can be understood as the analogues of the electric field intensity and potential in electrostatics. The lines of constants ρ and θ are orthogonal as electric lines and equipotentials are normal to one another. The mathematical meaning of ρ is the natural logarithm of the Jacobian for the conformal transformation w→z, as described by M. R Speigal, “Complex Variables”, Schaum's outline series, McGraw Hill, 1974. Hence ρ measures, in the logarithmic scale, the magnitude of the rate of change of z with respect to the rate of change of w. In other words, ρ0 measures, in logarithmic scale, the ratio of an elemental area, Δz, at z to its corresponding area, Δw, at w(z) under a conformal transformation z⇄w. Hence ρ0 evaluated at the concave parts of a surface is bigger than ρ0 at the convex parts of the surface. FIG. 4 shows the conformal co-ordinates graphically; the bold lines (24) are the constant v's and the normal lines (25) are the constant u's; points a (26), b (28) and c (30) represent the flat, convex and concave curvatures; ρ0 associate with these points are ordered as ρ0(c)>ρ0(a)>ρ0(b)
By means of a conformal transformation, Cauchy-Riemann condition and change of integration variables, the deformation energy per unit surface area per unit groove, g, can be expressed in the three different ways as follows:
                                          g            0                    =                                    K                              2                ⁢                L                                      ⁢                          ∫                              ∫                                                      ⅆ                    x                                    ⁢                                      ⅆ                    y                                    ⁢                                      {                                                                                            (                                                                                    ∂                              θ                                                                                      ∂                              x                                                                                )                                                2                                            +                                                                        (                                                                                    ∂                              θ                                                                                      ∂                              y                                                                                )                                                2                                                              }                                                                                      ⁢                                                      (                  36          ⁢          a                )                                          =                                    K                              2                ⁢                L                                      ⁢                                          ∫                0                ∞                            ⁢                                                ⅆ                  v                                ⁢                                                      ∫                    0                    L                                    ⁢                                                            ⅆ                      u                                        ⁢                                          {                                                                                                                                  ∂                              ρ                                                                                      ∂                              u                                                                                ⁢                                                                                    ∂                              θ                                                                                      ∂                              v                                                                                                      -                                                                                                            ∂                              ρ                                                                                      ∂                              v                                                                                ⁢                                                                                    ∂                              θ                                                                                      ∂                              u                                                                                                                          }                                                                                                          ⁢                                                      (                  36          ⁢          b                )                                          =                                    K                              2                ⁢                L                                      ⁢                          ∫                              ∫                                                      ⅆ                    ρ                                    ⁢                                      ⅆ                    θ                                                                                      ⁢                                                      (                  36          ⁢          c                )            where L is the pitch of the groove of a given grating and K=K11=K22=K33 is the elastic constant.
Point singularities can be added to the solution of equation 31 without altering the boundary condition requirement of homeotropic surface condition of equation 32. For a periodic grating surface, we consider an array of singularities on the surface. Hence, the tilt angle, θ, of the director field becomes:
                    θ        =                              Im            ⁢                                                  ⁢                          ln              ⁡                              (                                                      ⅆ                    z                                                        ⅆ                    w                                                  )                                              +                      π            2                    +                      Im            ⁢                                                  ⁢                                          ∑                α                N                            ⁢                                                          ⁢                                                ∑                                      n                    =                                          -                      ∞                                                        ∞                                ⁢                                                                  ⁢                                                      m                    α                                    ⁢                                      ln                    ⁡                                          (                                              w                        -                                                  λ                          α                                                -                                                  2                          ⁢                          π                          ⁢                                                                                                          ⁢                          n                                                                    )                                                                                                                              (        37        )            where mα and N are half integers representing the strength of the N nematic disclinations on a single groove of the grating surface, λα are N real numbers defined in the range (−π, π) representing the positions (w=u+i0) of the mα disclinations in the 0th groove of the grating. In the transformed w plane, the periodicity of the groove is conveniently scaled to 2π. The logarithmic functions in the summation in equation 37 are analogues of the well known functions for line charges of strength mα in two dimensional electrostatics.
Upon summing the infinite series in equation 37, on yields:
                              ρ          +                      i            ⁢                                                  ⁢            θ                          =                              ln            ⁢                                          ⅆ                w                                            ⅆ                z                                              +                                    ∑              α              N                        ⁢                                          m                α                            ⁢              ln              ⁢                                                          ⁢                              sin                ⁡                                  (                                                            w                      -                                              λ                        α                                                              2                                    )                                                              +                      i            ⁢                          π              2                                                          (        38        )            
The deformation energy for θ given solution 38 is:
                              g          d                =                              g            o                    +                                                    π                ⁢                                                                  ⁢                k                                            2                ⁢                                                                  ⁢                L                                      ⁢                          {                                                                    ∑                    α                    N                                    ⁢                                                            m                      α                                        ⁢                                          ρ                      ⁡                                              (                                                  λ                          α                                                )                                                                                            +                                                      ∑                                          α                      ,                      β                                        N                                    ⁢                                                            m                      α                                        ⁢                                          m                      β                                        ⁢                                          ln                      ⁡                                              [                                                                              sin                            2                                                    ⁡                                                      (                                                                                                                            λ                                  α                                                                -                                                                  λ                                  β                                                                                            2                                                        )                                                                          ]                                                                                                        }                                                          (        39        )            
The expression is valid for
            ∑      α        ⁢          m      α        =  0.Otherwise, g will increase linearly with the system height. In the second summation,
              λ      α        -          λ      β        is defined to be r when α=β. r<<L is the diameter of the disclination cores.
The second summation in equation 38 is always positive. The condition for the possibilities of multi-stable states is given by:
                                                        ∑              α              N                        ⁢                                          m                α                            ⁢                              ρ                ⁡                                  (                                      λ                    α                                    )                                                              +                                    ∑                              α                ,                β                            N                        ⁢                                          m                α                            ⁢                              m                β                            ⁢                              ln                ⁡                                  [                                                            sin                      2                                        ⁡                                          (                                                                                                    λ                            α                                                    -                                                      λ                            β                                                                          2                                            )                                                        ]                                                                    ≤        0                            (        40        )            for some configuration disclination lines on the grating position at λα's.
To minimise the first summation, the following criteria are observed:mα>0ρ(λα)=min(ρ(λ))  (41a)mα<0ρ(λα)=max(ρ(λ))  (41b)
It concludes that a positive indexed disclination (mα>0) tends to form at the position on a grating where the curvature is convex (ρ<c) and a negative index disclination (mα<0) tends to form at the position with a concave curvature (ρ>c).
For a given set of λα's, the tilt angle of the director field at a distance far from the grating surface can be found from equation 38 for θ by taking the limit of v to infinity:
                                          lim                          v              →              ∞                                ⁢          θ                =                              -                                          ∑                α                N                            ⁢                                                m                  α                                ⁡                                  (                                                            λ                      α                                        2                                    )                                                              +                      π            2                                              (        42        )            
Implementation of this theory, allows the energy associated with a particular position of a pair of +½ and −½ defects, and the pretilt imparted to the liquid crystal at an infinite distance from the surface, to be calculated for an arbitrary grating shape. An arbitrary grating shape is input into the model and the energy is calculated as a function of both −½ defect position along the grating and +½ defect position along the grating. At points where the −½ defect is coincident with the −½ defect, the defects annihilate and the non-defect state is formed.
FIG. 5 shows a series of gratings and a representation of a cross-section through the energy profile. The energy profile is obtained as a function of both −½ defect position and +½ defect position on the grating, producing a three dimensional energy profile. FIGS. 5b, 5d, and 5f show a cross section of this profile (in each case the same profile).
FIG. 5a represents a symmetrical sinusoidal grating. It can be seen that two energy minima occur (30, 31). The first energy minimum (30) is associated with coincident −½ and +½ defect positions (i.e. the non-defect state) has a calculated pretilt of 90°. The second energy minimum (31) is associated with the +½ defect being at the convex defect site and the −½ defect being at the concave defect site; this gives a pretilt of 0°.
FIG. 5c represents an asymmetric sinusoidal grating. It can be seen that two energy minima occur (33, 34). The first energy minimum (33) is associated with coincident −½ and +½ defect positions (i.e. the non-defect state) has a calculated pretilt of 90°. The second energy minimum (34) is associated with the +½ defect being at the convex defect site and the −½ defect being at the concave defect site; this gives a pretilt of 36°. Hence, as found for the simple model above, an asymmetric grating with defect sites not fulfilling the criteria a=L/2 produces a finite pretilt.
FIG. 5e represents an asymmetric square grating with four defect sites. It can be seen that five energy minima occur (35, 36, 37, 38, 39). The energy minimum (37) is associated with coincident −½ and +½ defect positions (i.e. the non-defect state) has a calculated pretilt of 90°. The other four energy minima (35, 36, 38, 39) are associated with the various combinations of +½ defect and −½ defect positions as described with reference to 7. The energy minimum (39) gives a pretilt of 0°.
To summarise, it can be seen that a computer implementation of the above model allows the surface pretilt induced by a surface alignment grating, and the energy associated with that particular configuration, to be calculated from a particular defect pair position on a given surface alignment grating profile.
Multiple Zenithally Stable States and the Associated Surface Pretilt.
The model described above with reference to FIGS. 4 and 5 demonstrates that for a given surface profile it is possible to obtain a plurality of stable surface states, and allows the pretilt associated with such states to be calculated. Such multi-stability is found within one period of the grating when there are three or more defect sites and at least one defect site has a maximum curvature which is convex and at least one defect site has a maximum radius of curvature which is concave.
A plurality of defect sites per unit length of grating allows the formation of defect pairs at a plurality of positions within a single grating period. It is only possible for +½ and −½ defects to form in pairs; a single defect, or two −½ defects, can not form alone. Each +½ and −½ defect pair position induces a liquid crystal pretilt that could be calculated rigorously using the model described with reference to FIGS. 4 and 5 above, or less rigorously using the simple model described with reference to FIGS. 1 to 3 above.
A plurality of defect sites per grating period, the formation of defect pair combinations and the pretilt associated with the various defect site configurations will now be described with reference to FIGS. 6 to 9.
FIG. 6 depicts a surface alignment grating (40) with a single +½ defect site (42) per grating period, L. The surface alignment grating has two possible −½ defect sites per grating period; a first −½ defect site (44) and a second −½ defect site (46). Two possible defect states could then be formed with a +½ defect at the +½ defect site (42) and a −½ defect at either the first −½ defect site (44) or the second −½ defect site (46). These two possible defect pair configurations are shown in FIGS. 6a and 6b, where representative −½ defects (48) and +½ defects (50) are also shown.
The symmetric grating of FIG. 6 can produce two possible pre-tilt angles θ1=πd1/L (for the configuration of FIG. 6a) and θ2=πd2/L (for the configuration of FIG. 6b), which are finite but symmetrical about the substrate normal. A non-defect configuration can also be adopted by the nematic liquid crystal at the grating surface. The surface alignment grating of FIG. 6 thus provides three (two defect and a non-defect) stable surface pretilt configurations.
Referring to FIG. 7, there exists on the surface alignment grating (52) a first +½ defect site (54), a second +½ defect site (56), a first −½ defect site (58) and a second −½ defect site (60). Liquid crystal (2) in contact with the surface alignment grating (52) could adopt any one of four possible defect configurations, as shown in FIGS. 7a, 7b, 7c and 7d. The defect pair configurations shown in FIGS. 7a, 7b, 7c and 7d will induce surface pretilts of θ1, θ2, θ3 and θ4 respectively. The surface pretilt angles θ1, θ2, θ3 and θ4 can be estimated using equation 17 from the ratios of distances d1, d2, d3, d4 to L.
For a symmetric grating of the type shown in FIG. 7, the surface pretilt angles θ1 and θ2 occupy angles which are complimentary, and are therefore symmetric about the substrate normal. The same is true for θ3 and θ4. In practice the θ1 and θ2 states have the lower energy configuration since the high average surface tilt possessed by θ3 and θ4 can be more readily attained by adopting the non-defect high tilt configuration. Equivalent but opposite pretilts may also have advantages, such as wider viewing angle (similar to dual domain TN).
Referring to FIG. 8, it is possible for a symmetric or an asymmetric grating to have a surface profile such that a certain combination, or certain combinations, of +½ defect sites and −½ defect sites will fulfil the zero pretilt criteria of equation 17 thus producing one or more defect states of zero surface pretilt (i.e. d=L/2).
FIG. 8a shows a symmetric alignment grating (62) which has four defect sites per unit length; a first −½ defect site (64), a second −½ defect site (66), a first +½ defect site (68) and a second +½ defect site (70). If a −½ defect forms at the first −½ defect site (64) and a +½ defect forms at the first +½ defect site (68), the criteria L=2d is fulfilled and the defect state will impart a zero pretilt to the liquid crystal (2) in contact with the grating surface (62). Similarly, zero pretilt is imparted to the liquid crystal (2) by the surface alignment grating (62) if a −½ defect forms at the second −½ defect site (66) and a +½ defect forms at the second +½ defect site (70). The two other possible combinations of the −½ and +½ defect position will impart a non-zero surface pretilt to the liquid crystal layer (2) according to equation 17.
FIG. 8b shows an asymmetric alignment grating (72) which has four defect sites per unit length; a first −½ defect site (74), a second −½ defect site (76), a first +½ defect site (78) and a second +½ defect site (80). If a −½ defect forms at the second −½ defect site (76) and a +½ defect forms at the second +½ defect site (80), the criteria L=2d is fulfilled and, when this particular defect state is formed, the grating surface (72) imparts a zero pretilt to the liquid crystal (2). The three other possible combinations of the −½ and +½ defect position will impart a non-zero surface pretilt to the liquid crystal layer (2) according to equation 17. Note that although there are four defect sites in this example, they alternate convex, convex, concave, concave and therefore an additional defect pair cannot be created.
Each of the five possible states that may form on the symmetric alignment grating of FIG. 8a, or the asymmetric grating of FIG. 8b, will induce a certain liquid crystal surface pretilt angle and will have correspondingly different energies. The lower energy configurations will be those which induce the lower pretilt angles. If an asymmetric grating was so profiled, with an appropriate groove depth and pitch, the energy of one or more of the defect states could be made such that those defect states are less energetically favourable than the non-defect state and some or all of the other defect states.
Surface grating structures can be designed, using the models and examples described above, where the defect sites are positioned so as to induce a plurality of states of various different energies, and/or where the surface pretilt of one or more of the defect states is to be controlled (for example to get defect states which induces substantially zero pretilt).
The construction of multi-faceted gratings is limited only by the fabrication process. However it is unlikely that a grating with more than approximately 10 defect sites per grating period would be required. FIG. 9 shows an alignment grating (82) with four −½ defect sites (84, 86, 88, 90) per unit length and four +½ defect sites (92, 94, 96, 98) per unit length. In addition to the non-defect (homeotropic) state, the alignment grating structure (82) would enable sixteen defect state configurations to be formed. No more than one pair of defect can form a defect state on this surface; to do so requires alternating +½ and −½ defect sites. The surface pretilt imparted to the liquid crystal by each of the defect states is defined by equation 17, and could also be calculated using the more rigorous model described above with reference to FIGS. 4 and 5.
Surface alignment gratings may be profiled so as to favour the formation of more than one pair of +½ and −½ defects per unit length of grating, by producing structures with alternating +½ and −½ defect sites.
An example of a surface alignment grating, profiled so as to favour the formation of more than one pair of +½ and −½ defects per unit length of grating, will now be described with reference to FIG. 10. The alignment imparted to a nematic liquid crystal layer by the type of grating structures described with reference to FIG. 10 will then be described with reference to FIGS. 11 to 12.
A blazed surface alignment grating (100) of pitch P and amplitude A is shown in FIG. 10a, and may be described by the function:
                              y          ⁡                      (            x            )                          =                              f            ⁢                                                  ⁢                          n1              ⁡                              (                x                )                                              =                                    A              2                        ⁢                          sin              ⁡                              (                                                                            2                      ⁢                                                                                          ⁢                      π                      ⁢                                                                                          ⁢                      x                                        p                                    +                                      δ                    ⁢                                                                                  ⁢                                          sin                      ⁡                                              (                                                                              2                            ⁢                                                                                                                  ⁢                            π                            ⁢                                                                                                                  ⁢                            x                                                    p                                                )                                                                                            )                                                                        (        43        )            where: A=half of the peak to peak amplitude of the grooves                p=period or pitch of the grooves        δ=the asymmetry of the function (δ=0 gives a sinusoidal function and δ>0 gives a blazed asymmetric profile).        
Within an overall period p, it is possible to have two or more sub-periods which are, for example, of pitch p1 and p2 and of amplitude A1 and A2. Such surface profiles will now be described for the sinusoidal function ƒn1(x), described by equation 43 above, with reference to FIGS. 10b to 10d. The technique of building up an overall surface alignment grating of period p from two smaller sub-units of a smaller pitch and/or of different amplitudes is not restricted to sinusoidal functions of the form described herein but may also be applied to a plurality of sub-units provided that the condition p1+p2 . . . +pn=p is met.
FIG. 10b shows a surface alignment grating (102) made up of one sub-period of ƒn1(x) (given in equation 43) with pitch p1 and amplitude A, plus one sub-period of ƒn1(x) with pitch p2 and amplitude A, such that p=p1+p2. In FIG. 10b, p2>p1. FIG. 10c shows a surface alignment grating (103) profiled such that one period p of the grooved surface is made up of one sub-period of the function ƒn1(x) with pitch p/2 and amplitude A1 plus one sub-period of the function ƒn1(x) with pitch p/2 and amplitude A2. FIG. 10d shows a surface alignment grating (120) profiled such that one period p of the grooved surface is made up of one period of the function ƒn1(x) with pitch p1 and amplitude A1 plus one period of the function ƒn1(x) with pitch p2 and amplitude A2.
The surface alignment grating structure (102) given in FIG. 10b, when in contact with a nematic liquid crystal (2) is depicted in FIG. 11. The nematic liquid crystal director, n, is perpendicular to the iso-contour lines (101). From FIG. 11, it can be seen that the surface alignment grating structure (102) has a first (104) and second (106) +½ defect site and a first (108) and second (110) −½ defect site. A non-defect, homeotropic, state is depicted in FIG. 11a. 
In FIG. 11b, a +½ defect (112) forms at the first +½ defect site (104) and a −½ defect (114) forms at the first −½ defect site (108), giving an intermediate pretilt state governed by the relative position of the defect pair according to equation 17 as described above.
In FIG. 11c, a +½ defect (112) forms at the first +½ defect site (104) and a −½ defect (114) forms at the first −½ defect site (108) and in addition a second +½ defect (116) forms at the second +½ defect site (106) and a second −½ defect (118) forms at second −½ defect site (110). The formation of the two defect pairs per unit length of grating will produce a low surface pretilt state wherein the pretilt adopted by the liquid crystal a short distance, compared with the cell gap, from the grating surface (102) will be the average of that expected (from equation 17 above) for the defect states of a grating of pitch p1 and a grating of pitch p2. The different grating profiles of the portion of the grating of pitch p1 and the portion of the grating of pitch p2 causes the first and second defect pairs to have different energies associated with their formation. The energy associated with a defect pair on a grating surface can be calculated using the detailed model of grating induced surface pretilt described above with reference to FIGS. 4 and 5. Qualitatively, it can be seen that the different groove depth (A) to pitch (p) ratio of the two sub-components of the surface alignment grating (102) will be different, and consequently that the energy associated with the formation of defect pairs on the two sub-components of the surface alignment grating (102) will also differ.
In summary, it can be seen from FIG. 11 that there are three possible configurations, each producing a different surface pretilt angle, that are adopted when two pairs of defects form on two sub-periods of a grating; namely a non-defect, a single defect pair state and a dual defect pair state. The single defect pair state could give rise to other variants, but these are less likely to occur because the energy of formation associated with sharp facets is significantly different to that associated with the shallower facets.
The surface alignment grating structure (120) given in FIG. 10d, when in contact with a nematic liquid crystal (2) is depicted in FIG. 12. Referring to FIG. 12, it can be seen that a similar behaviour to that described with reference to FIG. 11 is obtained with a surface alignment grating (120) of an overall period p that consists of two sub-periods p1 and p2 with corresponding amplitudes A1 and A2. The surface alignment grating (120) has a first (122) and second (124) +½ defect site and a first (126) and second (128) −½ defect site. A non-defect, homeotropic, state is depicted in FIG. 12a. 
From FIG. 12b a defect state is shown where a first +½ defect (130) forms at the first +½ defect site (122) and a first −½ defect (132) forms at the first −½ defect site (126), giving an intermediate pretilt state governed by the relative position of the defect pair according to equation 17 as described above. FIG. 12c depicts a defect state wherein two defect pairs have formed; a first +½ defect (130) at the first +½ defect site (122), a first −½ defect (132) at the first −½ defect site (126), a second +½ defect (134) at the second +½ defect site (124) and a second −½ defect (136) at the second −½ defect site (128). As described with reference to FIG. 11 above, the pretilt associated with the defect state of FIG. 12b is governed by equation 17 whereas the pretilt associated with the defect pair state of FIG. 12c can be calculated by determining the average pretilt of two gratings of pitches p1 and p2. The energy associated with each defect pair can, as described with reference to FIG. 11 above, be determined using the model described herein with reference to FIG. 4 and 5.
To summarise, the surface alignment grating of period P described with reference to FIG. 12 above contains two distinct sub-period regions. The difference in defect energy associated with each of the sub-periods means that it is possible for stable states to form with no defects, one defect pair or two defect pairs. Each of these states will produce a different surface pretilt, allowing a tri-stable device to be readily constructed from such a surface alignment grating
As described above with reference to FIG. 10 to 12, a grating consist of a number of sub-periods. It is also possible for each sub-period to possess more than two defects sites, thereby combining the structures described in FIGS. 6 to 9 in the manner described with respect to FIGS. 10 to 12.
To achieve multi-stability, i.e. a surface alignment grating which imparts three or more stable pretilt angles in the same azimuthal plane to the liquid crystal in the vicinity of the surface, when more than one pair of defects form per unit length of grating requires the overall grating period p to be less that half of the gap between the upper and lower cell walls. If the overall pitch p is more than approximately half the cell gap, defect pairs associated with adjacent sub-unit grating periods contained within the overall grating period will induce a surface pretilt that does not, in the vicinity of the surface, combine so as to form a plurality of stable pretilt angles in the same azimuthal plane. If the overall pitch p is more than approximately half the cell gap and, for example, two defect pairs per grating period could form, the surface pretilt associated with each defect pair would not merge within a short distance of the surface but would produce two distinct and adjacent regions of different bulk liquid crystal configuration (i.e. a striped texture when observed optically).
Although achieving a uniform pretilt within a short distance of the surface alignment grating, thereby ensuring that a uniform optical texture is attained, is ideal the formation of adjacent regions of different bulk liquid crystal configuration (thus adjacent regions of different optical appearance) is perfectly acceptable for display applications provided that the size of such regions is sufficiently small so that the separate regions can not be readily perceived by display observers.
Device Configurations.
A person skilled in the art would be aware of numerous device configurations which would allow the multi-stable surface described herein to be exploited. Several of the possible liquid crystal cell and device configurations will now be described with reference to FIG. 13.
One cell configuration which allows the existence of a plurality of stable surface pre-tilt states to be exploited as a plurality of greyscale levels is shown in cross section in a stylised form in FIG. 13. The cell is constructed from two cell walls (140,142), where the first cell wall (142) has a multi-stable surface alignment grating (144) formed on its internal surface, in this example a tri-stable surface alignment grating in accordance with the teaching described above, whilst the second cell wall (142) has is treated so as to induce hometropic alignment to the nematic liquid crystal (2).
The first (142) and second (140) cell walls are maintained typically 1–10 μm apart by a spacer ring (not shown), numerous beads of the same dimension dispersed within the liquid crystal (not shown) or numerous beads of the same dimension dispersed within any glue used to bond the cell walls together (not shown). Many other techniques of maintaining a gap between the cell walls are readily known to a person skilled in the art.
A nematic liquid crystal (2) sandwiched between the first (142) and second (140) cell walls of can exist in any of three stable configurations; the non-defect configuration shown in FIG. 13(a), the defect configuration shown in FIG. 13(b) or the defect configuration shown in FIG. 13(c). The two defect state configurations of FIGS. 13(a) and 13(b) arising from two possible positions which the bend defect can adopt on a the tri-stable surface alignment grating (144). For many nematic materials, a splay or bend deformation will lead to a macroscopic flexoelectric polarisation which is represented by the vector P in FIG. 13(b) and P′ in FIG. 13(c). A dc pulse applied to the electrodes (146) can couple to this flexoelectric polarisation, which is predominantly close to the alignment surface, and depending on its sign and magnitude any one of the three configurations can be selected.
The cell configuration used to exploit the existence of a tri-stable surface alignment grating, described with reference to FIG. 13 above, will also allow multi-stable surface alignment gratings to be exploited as multi-stable devices. Maximum contrast will be achieved if one of the defect states adopts a zero pretilt configuration by having defects positioned so as to fulfil the criteria a=L/2.
To obtain a display device with optical contrast, the cell configurations described above can be combined with external polariser(s) and/or a reflector and the device may be operated in either transmissive or reflective mode. Alternatively, a dichroic dye may be mixed with the liquid crystal to get different absorption of light in the various multi-stable configurations. All these techniques of producing optical contrast from liquid crystal cell configuration are well known to a person skilled in the art.
Optical compensation techniques, which would be well known to person skilled in the art, may also be employed to enhance the optical contrast and viewing angle characteristics of any of the plurality of defect states.
Any nematic material with positive, negative or zero dielectric anisotropy may be used in the cell configuration. Dual frequency nematic materials, where the sign of dielectric anisotropy changes with the frequency of applied voltage, may also be used. In a preferred embodiment a high positive dielectric anisotropy, and associated high flexoelectric coefficient, may be used. Super-twist nematic and active matrix nematic materials would be suitable, although there is no requirement for the high ionic purity of liquid crystal required for the operation of active matrix devices.
The cell walls may be formed of a relatively thick non flexible material such as a glass, or one or both walls may be formed of a flexible material such as a thin layer of glass or a plastic flexible material e.g. polyolefin or polypropylene. A plastic cell wall may be embossed on its inner surface to provide a grating. Additionally, the embossing may provide small pillars (e.g. of 1–10 μm height and 5–50 μm or more width) for assisting in correct spacing apart of the cell walls and also for a barrier to liquid crystal material flow when the cell is flexed. Alternatively the pillars may be formed by the material of the alignment layers.
The grating may be a profiled layer of a photopolymer formed by a photolithographic process e.g. M C Huntly, Diffraction Gratings (Academic Press, London, 1982) p 95–125; and F Horn, Physics World, 33 (March 1993). Alternatively, the grating may be formed by embossing; M T Gale, J Kane and K Knop, J App. Photo Eng, 4, 2, 41 (1978), or ruling; E G Loewen and R S Wiley, Proc SPIE, 88 (1987), or by transfer from a carrier layer. A grating of 0.2 μm to 5 μm pitch, with a groove depth of between 0.1 μm to 3 μm is preferred.
Experimental examples, to backup the theoretical analysis described above, will now be described with reference to FIGS. 14 to 22.